Belgikar 2024 new applications ergodic theory sets differences

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Kabir Belgikar, Vitaly Bergelson, Gabriel Black, David Kruzel, New Applications of Ergodic Theory to Sets of Differences. arXiv preprint (2024). arXiv:2412.01185. The copy read for this card is arXiv:2412.01185v2 (19 July 2025), and the theorem labels below are that version's.

The paper gives a short ergodic proof of Ruzsa's theorem (their Theorem 1.1: for S_1,...,S_k of positive upper density there is S with d(S) >= prod dbar(S_i) and Delta_1(S) contained in the intersection of the Delta_2(S_i), with that intersection syndetic) and then extends it in two directions. Theorem 1.2 handles the sparsified difference sets Delta_{1,c} and Delta_{2,c} defined via [n^c] for non-integer c > 0, keeping the sharp bound d(S) >= prod of the upper densities and showing the intersection D_c is thick; Theorem 1.3 covers integer c with d(S) > 0 and D_c syndetic. Theorem 1.5 (proved as Theorem 4.9) generalizes the whole statement to a countably infinite amenable group with arbitrary left Folner sequences and one left tempered sequence, and Theorem 4.24 extends it to cancellative amenable semigroups; Section 5 gives tempered Folner sequences in (N,+), (N,times), the Heisenberg group and locally finite groups. The engine is pointwise ergodic theory, in particular Lindenstrauss's pointwise theorem for tempered Folner sequences, together with a Poincare-recurrence extension. For problem 332 this confirms a genuine math.DS ergodic-theory generalization of the classical Stewart-Tijdeman/Ruzsa sufficient condition. Its Theorem 4.1, which allows any Folner sequence, applied along intervals on which a set has density tending to its upper Banach density, gives bounded gaps for the infinite-difference set of every set of positive upper Banach density, the class that the note linked from the Problem 332 thread on 4 May 2026 claims.

Source: https://arxiv.org/abs/2412.01185. The arXiv record (https://arxiv.org/abs/2412.01185, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #332

Results to transcribe.

• Theorem 1.1 (Ruzsa): Restated classical result: for S_i of positive upper density there is S with d(S) >= prod dbar(S_i), Delta_1(S) inside the intersection D of the Delta_2(S_i), and D syndetic with at most prod 1/dbar(S_i) translates covering Z.
• Theorem 1.2: For non-integer c > 0, the analog with Delta_{1,c}, Delta_{2,c} holds with d(S) >= prod of upper densities and D_c thick.
• Theorem 1.3: For integer c, the same containment holds with d(S) > 0 and D_c syndetic (but possibly not thick).
• Theorem 1.5 / Theorem 4.9: Amenable-group version: for a countably infinite amenable group with left Folner sequences G (tempered), F_1,...,F_k, there is S with d_G(S) >= prod of the upper F_i-densities and Delta_1(S) inside the intersection of the Delta_2(F_i,S_i), which is syndetic.
• Theorem 4.24: Extension of Theorem 4.9 to countably infinite cancellative amenable semigroups via embedding into the group of quotients; there the right translates D m_i^{-1} cover G, and the left translates too when G is a group.